The Hartle-Hawking-Israel state on stationary black hole spacetimes

We consider a free quantized Klein-Gordon field in a spacetime (M, g) containing a stationary black hole, more precisely a spacetime with a stationary bifurcate Killing horizon in the sense of Kay and Wald. We prove the existence of the Hartle-Hawking-Israel ground state, which is a pure state on the whole spacetime whose restriction to the exterior of the black hole is a thermal state at Hawking temperature.We show that the HHI state is a Hadamard state and is the unique Hadamard extension of the above thermal state to the whole spacetime. We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderon projector for elliptic boundary value problems.